L-function 2-40e2-8.5-c1-0-32

• Label: 2-40e2-8.5-c1-0-32
• Degree: $2$
• Conductor: $1600$
• Sign: $-0.707 + 0.707i$
• Analytic cond.: $12.7760$
• Root an. cond.: $3.57436$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 2i·3^-s − 9^-s − 6i·11^-s + 6·17^-s + 2i·19^-s − 4i·27^-s − 12·33^-s − 6·41^-s − 10i·43^-s − 7·49^-s − 12i·51^-s + 4·57^-s − 6i·59^-s + 14i·67^-s − 2·73^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$1600$    =    $2^{6} \cdot 5^{2}$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (801, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.593109345$
$L(\frac{3}{2})$		not available

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
good	3	$1 + 2iT - 3T^{2}$
	7	$1 + 7T^{2}$
	11	$1 + 6iT - 11T^{2}$
	13	$1 - 13T^{2}$
	17	$1 - 6T + 17T^{2}$
	19	$1 - 2iT - 19T^{2}$
	23	$1 + 23T^{2}$
	29	$1 - 29T^{2}$
	31	$1 + 31T^{2}$

Imaginary part of the first few zeros on the critical line

−8.851616209863958700260998977023, −8.172631021432919553864616120885, −7.60788838758082033607254322932, −6.68281195308945160314481488920, −5.95307842161694148323001762585, −5.28142301435937642599599931765, −3.77096149761736834639654490818, −2.98866927850474552144855626409, −1.65869642098408123024634693077, −0.65002429655742763620723412684, 1.58120374473771042556784119673, 2.94778991351403219971777831753, 3.92857836024839309430758945156, 4.74599182667682973572692993289, 5.26516734667149971724317250711, 6.50593193379168764596163133492, 7.37550334194041508567538602736, 8.103774700995373890607445658339, 9.313892019799464381556970285465, 9.694416774443088210999715477646

Graph of the $Z$-function along the critical line